Finding the Least Number with a Specific Remainder

The problem asks for the least number which, when divided by 12, 20, and 24, leaves a remainder of 8 in each case. To solve this type of problem, we first need to find the least common multiple (LCM) of the divisors (12, 20, and 24).

Understanding the Concept

If a number leaves the same remainder 'r' when divided by several numbers, say a, b, and c, then the number minus 'r' must be exactly divisible by a, b, and c. This means that the number minus 'r' is a common multiple of a, b, and c. The least such number minus 'r' is the LCM of a, b, and c. Therefore, the least number itself is the LCM of a, b, and c, plus the remainder 'r'.

In this specific problem, the divisors are 12, 20, and 24, and the remainder is 8.

Step 1: Find the LCM of 12, 20, and 24

We can find the LCM by using the prime factorization method.

• Prime factorization of 12: $$12 = 2 \times 2 \times 3 = 2^2 \times 3^1$$
• Prime factorization of 20: $$20 = 2 \times 2 \times 5 = 2^2 \times 5^1$$
• Prime factorization of 24: $$24 = 2 \times 2 \times 2 \times 3 = 2^3 \times 3^1$$

To find the LCM, we take the highest power of all the prime factors that appear in any of the factorizations:

• Highest power of 2 is $2^3$ (from 24)
• Highest power of 3 is $3^1$ (from 12 and 24)
• Highest power of 5 is $5^1$ (from 20)

LCM(12, 20, 24) = $2^3 \times 3^1 \times 5^1 = 8 \times 3 \times 5 = 120$.

Step 2: Calculate the Required Number

The least number that leaves a remainder of 8 when divided by 12, 20, and 24 is given by the formula:

Least Number = LCM(12, 20, 24) + Remainder

Least Number = $120 + 8 = 128$.

Verification

Let's check if dividing 128 by 12, 20, and 24 leaves a remainder of 8:

• $128 \div 12$: $128 = 12 \times 10 + 8$ (Remainder is 8)
• $128 \div 20$: $128 = 20 \times 6 + 8$ (Remainder is 8)
• $128 \div 24$: $128 = 24 \times 5 + 8$ (Remainder is 8)

Since the remainder is 8 in all cases, 128 is indeed the correct number. As we used the LCM, it is the least such positive number.

Thus, the least number which when divided by 12, 20 and 24 leaves in each case a remainder of 8 is 128.

Revision Table: LCM and Remainder Problems

Additional Information: Related Concepts

What is LCM?

The Least Common Multiple (LCM) of two or more numbers is the smallest positive integer that is a multiple of all the given numbers. It is useful in problems involving events that repeat at regular intervals or in finding the least number that is exactly divisible by a set of numbers.

Finding LCM by Division Method:

Another way to find the LCM is using the division method:

LCM = $2 \times 2 \times 3 \times 5 \times 2 = 120$.

Both methods yield the same LCM. Understanding LCM is crucial for solving problems involving finding numbers that meet specific divisibility and remainder conditions.